INTRODUCTION

From one point of view, the present paper is mainly concerned with specialising the results on the groups J(X), given in previous papers of this series, to the case X = Sn. It can, however, be read independently of the previous papers in this series; because from another point of view, it is concerned with the use of extraordinary cohomology theories to define invariants of homotopy classes of maps; and this machinery can be set up independently of the previous papers in this series. We refer to them only for certain key results.

From a third point of view, this paper represents a very belated attempt to honour the following two sentences in an earlier paper. “However, it appears to the author that one can obtain much better results on the J-homomorphism by using the methods, rather than the results, of the present paper. On these grounds, it seems best to postpone discussion of the J-homomorphism to a subsequent paper.” I offer topologists in general my sincere apologies for my long delay in writing up results which mostly date from 1961/62.

I will now summarise the results which relate to the homotopy groups of spheres. For this one needs some notation.